Exact deflation in the complex modal analysis of low-rank nonclassically damped structures

This paper presents the complex modal analysis for a proportionally damped structure equipped with linear non-proportionally damped viscous elements (substructures or discrete real devices) giving a low-rank contribution (r) to the non-proportional part of the damping matrix. Using the classical undamped modes and a special low-rank matrix update formulation of the original problem, the original Quadratic Eigenproblem (QEP) is hugely deflated, without approximations, to an equivalent Rational Eigenproblem (REP) of dimension r << n (Theorem 2), as an alternative to the linearized Standard Eigenproblem of order 2n over the complex field. The existence of classical modes in non-classically damped structures is also discussed. The REP is solved by the homotopy method: a robust predictor-corrector continuation algorithm is designed in order to determine the required eigenpairs. Some application to simple models of both traditional and base-isolated structures, together with an outline of future work, end the paper.